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Sunday, February 03, 2008

M Theory Lesson 153

On the pair of pants, the hexagon on one side is divided into three pentagons. This triple is associated to the three level quantum number, called mass. Recall that these become three heptagons on a tiling of the Klein quartic. There are 24 such tiles.

Observe how the so-called higher dimensions of string theory are simply spaces spanned by quantum numbers: two for spin, three for mass, six for em charge and so on. In this light we can reevaluate ridiculous expressions of the form, "gravitons leak into extra dimensions". If a graviton is a particle, either it has certain quantum numbers or it doesn't. It cannot decide to take a leak and find itself more quantum numbers.

The 24 tiles are the 24 dimensions of the Leech lattice. The 8 hexagons are the 8 dimensions of the E8 lattice. Each of these exhibits a triality. I'm beginning to find this quite a lot of fun.

16 Comments:

Another step in making sense of string theory might be to see that strings have a natural physical interpretation as World-Lines of particles, with open strings representing particles that enter and leave (asymptotically) from a Feynman-diagram-type event, and closed strings representing loops in a Feynman-diagram-type event.

I don't understand why there is so much resistance to that viewpoint. For example, I mentioned it by e-mail to Michael Atiyah in connection with his 2005 talk at KITP Santa Barbara in which he described a class of physics models based on "... the past history of a particle moving as a real particle …", and he replied "... Yes my idea is to make things depend on the past world line of the particle. ... No, I do not think world lines are strings. ...".

Thanks for the comment, Tony. The way I see it, strings were just not set up that way, so I wouldn't call your interpretation 'strings' at all, even if your worldline matches a string perfectly in some mathematical realm. 'String' is a physical term, and if one wants to discuss completely different physics, it might be best to make up new terms. I realise this is not your style, and I do understand that communicating physics in new terms can be a bit challenging!

"Observe how the so-called higher dimensions of string theory are simply spaces spanned by quantum numbers: two for spin, three for mass, six for em charge and so on."

Sorry, I can't figure out what you're talking about here. The "three quantum numbers for mass", for example - are you talking about familiar three-dimensional space? But then shouldn't you say that (e.g.) for a superstring, there are nine such numbers? And then, if you say that em degrees of freedom come from Kaluza-Klein, you're using degrees of freedom from the higher-dimensional metric tensor, and there are many more than nine (or ten) numbers in that case. So I have no idea what you're talking about.

'Sure, there are all sorts of interesting ways of thinking about particle physics models using more “dimensions” than four. I would claim that the standard model is best thought of by thinking about a 16 dimensional space (a fiber bundle with fibers SU(3)xSU(2)xU(1) over spacetime). The thing for which there is no evidence is not extra dimensions in general, but extra Riemannian geometry dimensions where the metric now carries many more degrees of freedom with supposedly the same dynamics as the four we know.'

According to the Kaluza-Klein method of adding dimensions to incorporate gauge equations into general relativity, 1 dimension is added for Maxwell's equations which are interpreted by the mainstream as a simple U(1) Abelian symmetry (I won't mention my objection to this U(1) model here), while isospin charge represented by SU(2) requires 2 dimensions, and colour charge represented by SU(3) requires 4 extra dimensions. Adding 4 spacetime dimensions gives a total of 11 dimensions.

Lunsford's paper ‘Gravitation and Electrodynamics over SO(3,3)’ is a very simple and clear analysis which convincingly shows why the Kaluza-Klein 5 dimensional theory is a failure, and replaces it with a theory that has 3 time and 3 spatial dimensions.

The 3 spatial dimensions are associated with mass, making mass 3 dimensional, while the full em and gravity unification requires 6 dimensions altogether.

In my understanding, the 3 time dimensions are always identical, which has caused confusion up to now. Time dimensions are related to spatial dimensions by x = ct(x), y = ct(y), and z = ct(z), where t(x), t(y) and t(z) are all identical sized time dimensions.

You can see this in the fact that the Hubble expansion rate is the same in all directions. After 1998, it has been known that the universe isn't decelerating due to gravity as expected, so the ultimate measure of time is the age of the universe,

It's precisely because the expansion of the universe is isotropic (independent of the direction or axes), that time itself appears to be only 1 dimensional instead of 3 dimensional.

If you got 3 different values of the Hubble constant when measuring it from recession in 3 orthagonal directions using v = Hr, you'd get 3 different ages of the universe given by applying t = 1/H to the 3 different values of H.

It's because the expansion is isotropic that time itself seems to 1-dimensional. This is purely down to the fact that all the 3 time dimensions are indistinguishable!

My argument above isn't in Lunsford's paper, and his argument is purely abstract; using 3 time dimensions and 3 spatial dimensions you can unify general relativity and electromagnetism making a prediction. The prediction is that the cosmological constant is zero. This fits the observations if you do a non-ad hoc quantum gravity analysis, instead of forcing general relativity on to observations of supernovae redshift by adding an ad hoc small cosmological constant.

The quantum gravity analysis is that gravitons exchanged between receding masses over immense distances are redshifted, thus losing energy when received (this is described by Planck's law, relating the received energy of quanta to their received frequency). Such a redshift thus causes an energy loss, which must reduce the effective gravitational force coupling constant, G.

This loss means that distant receding objects are not slowed as much as predicted by GR. The mainstream GR cosmological recession model is basically classical Newtonian gravitation: the receding supernova at radial distance R is like a bullet fired upward from the earth being slowed by gravity (you just need to insert the supernova mass to replace the bullet's mass, and the mass of the universe contained within radius R to replace Earth's mass). This GR/Newtonian model of cosmology is false because it predicts an amount of gravitational deceleration of receding supernovas that is too large, due to the error in GR of assuming G is constant. When you reduce G in direct proportion to the redshift-induced frequency change factor for the distance of teh supernova, then you correct Perlmutter's results without needing to add a small cosmological constant. (There are also other small modifications suggested by quantum gravity, according to my research.)

Thus, Lunsford's 6-d unification correctly predicts that there is no cosmological constant (dark energy which is supposed to be speeding up the expansion to offset the unobserved gravitational deceleration predicted by the faulty mainstream GR model, which omits graviton redshift effects which reduce G).

Nobel Laureate Philip Anderson grasps the basic point about the cosmological constant hoax:

Mitchell, I am not referring to any classical spaces. Quantum numbers arise prior to any notion of spacetime, which is a highly derived concept. I might draw a picture of a circle, but it is more likely to represent logical set containment, as in a Venn diagram, than an objective component of spatial reality.

Kea, I can accept the proposition that spatial coordinates and quantum numbers have some sort of relationship. For example, you could think of it in terms of a change of basis in Hilbert space, from position eigenstates to energy eigenstates. What I don't understand is the specific assertion you made about extra dimensions in string theory and their relationship to spin, mass, and charge quantum numbers. I just can't make sense of it. Nor does the remark about gravitons make much sense. You have gravitons bound to a brane, but then sometimes they escape. You might be able to describe that in terms of a quantum number (perhaps momentum perpendicular to the brane) changing from zero to nonzero. I don't see the problem.

Nige, suppose the expansion of the universe was observably anisotropic. What would that have to do with extra time dimensions? The expansion would have been measured in three spatial dimensions, but against the same time dimension.

In purely algebraic approach one obtains very general definition of quantum numbers. Find the Cartan algebra of algebra in question and a basis of the algebra with well defined quantum numbers with respect to commutator action. You can also construct more general representations of the algebra. This allows all kinds of algebras and very general notion of quantum number of which Lie algebra based is only special case.

The conservative approach to dimensions and quantum numbers relies in the introduction of the notion of geometry as something obeying some symmetries and replacing algebras with Lie algebras.

The isometry group of higher-D space defines non-broken symmetries and vielbein group the broken symmetries for which there are no Noether charges.

If one accepts this picture, standard model quantum numbers leave just one option: H= M^4xCP_2. The introduction of induced spinor structure (after these 28 years still something new for the mainstream!) implies also two conserved fermion numbers corresponding to H-chiralities identifiable as quark and lepton number.

There are also topological quantum numbers: the most obvious ones being the genus for 2-D boundary components of 3-surfaces in case that one is ready to replaced strings with 3-D surfaces: the interpretation would be in terms of family replication.

I find it difficult to identify strings with orbits of particles. If one takes timelike string-orbit of particle- as basic object and applies stringy quantization one ends up with functional integral over 2-surfaces with ordinary time development replaced with evolution in space-like direction so that nothing new is obtained.

"Nige, suppose the expansion of the universe was observably anisotropic. What would that have to do with extra time dimensions? The expansion would have been measured in three spatial dimensions, but against the same time dimension." - mitchell

To answer your first point, "suppose the expansion of the universe was observably anisotropic. What would that have to do with extra time dimensions?", if the expansion of the universe was observably anisotropic, I'll explain how we'd be subjected to a net gravitational field from the surrounding universe, which would affect time (in fact we are subjected to this, but to a very small extent which is downplayed for obvious reasons, which was revealed by the CMB - it is however a way bigger anisotropy in the CMB than the small scale features discovered by COBE and WMAP, see http://adsabs.harvard.edu/abs/1978SciAm.238...64M).

The mechanism for how time is related to expansion is the gravitational field. The rate at which time flows depends on gravity, as is well known from experiments with atomic clocks. The stronger the gravitational field, the slower a clock runs.

The gravitational interaction is mediated by the exchange of gravitons between masses. A gravitational field is a flux of gravitons flowing between masses or locations containing energy.

If the expansion rate of the universe is bigger in one direction than in another, the mass will be distributed to greater distances where the expansion rate is bigger.

This means that the gravitational coupling constant will be weaker for interaction with the immense receding masses of material in directions where the recession rate is biggest (the gravitons will be more redshifted, to lower energy, when emitted by material which is receding faster).

Hence, if the universe is anisotropic, a clock will be subjected to a net gravitational field from the surrounding masses in the universe, which will affect time.

Secondly, if the universe were observably anisotropic, the age of the universe would vary according to the direction you looked. This would make the extra time dimensions observable.

The fact that the universe is anisotropic, does mean that the effective age of the universe is different in different directions! The Hubble constant does vary with direction slightly, but that is currently either ascribed to an "aether" (!) as in http://adsabs.harvard.edu/abs/1978SciAm.238...64M, or it is not analysed at all (glossed over).

There is no evidence that the anisotropy is the failure of special relativity, or proof of an "aether" by requiring that we have an absolute motion in the universe. The anisotropy is really evidence that there are 3 time dimensions, one corresponding to each spatial dimension.

The age of universe, i.e. time in cosmology, is t = 1/H, where Hubble constant H = v/R (here R is distance in a spatial dimension, and v is recession velocity). The age of the universe is calculated from the expansion rate, so your statement that:

"The expansion would have been measured in three spatial dimensions, but against the same time dimension",

is missing the point I'm making that different time dimensions emerge if you have an anisotropic expansion of the universe, because the Hubble expansion rate v = HR will be different in different spatial dimensions R, and so the measure of time as age of the universe t = 1/H, will be different for different spatial dimensions.

The only way to allow 3 spatial dimensions to give rise to 3 different ages of the universe is to increase the number of time dimensions until it coincides with the number of spatial dimensions.

An anisotropic expansion would mean that the Hubble constant varied with direction.

So you'd have a different age of the universe in different directions, as calculated from t = 1/H = 1/(v/R) = R/v.

The reason why there are 3 spatial dimensions is that we see an asymmetry when looking in different directions. If everything was the same in every direction, the fact that there are 3 spatial dimensions would be just as much an obscure mathematical finding as Lunsford's finding that there are 3 effective time dimensions.

Nige, earlier you wrote: "In my understanding, the 3 time dimensions are always identical, which has caused confusion up to now. Time dimensions are related to spatial dimensions by x = ct(x), y = ct(y), and z = ct(z), where t(x), t(y) and t(z) are all identical sized time dimensions."

That has to be a misunderstanding of what Lunsford, at least, is on about. Rather than saying that his theory has three time dimensions, it would make more sense to say that it has "three-dimensional time" - whatever that could mean! He talks about SO(3,3). There is no particular pairing up of space directions and time directions.

If what you were saying was the key to interpreting physical theories with multiple time dimensions, then any anisotropic physical process would indicate multiple times. But in fact it's just a matter of having clocks that run at different speeds - that's all that's going on in the case of anisotropic cosmic expansion! If you have one clock that runs twice as fast as another clock, you end up with two different coordinatizations of time, that differ by a scale factor, but it's still the same time dimension being measured.

Actually, I have never seen an explanation of how to interpret multiple time dimensions as time in the sense that we know it. Yes, you can formally generalize from the +++- signature of Minkowski space to ++-- or ++++-- or +++--- or whatever takes your fancy; but in so doing, the ability to interpret the timelike aspect of that metric as being about temporal order disappears.

mitchell says that he has "... never seen an explanation of how to interpret multiple time dimensions as time in the sense that we know it ...".

Itzhak Bars of Southern Cal has a lot of material about 2-time physics on his web page at http://physics1.usc.edu/%7Ebars/research.html#2THe says "... there is more to space-time than can be garnered with 1T-physics. 2T-physics introduces additional one space and one time dimensions ... 2T-physics tells us that the description of dynamics via the usual 1T-formalism should be interpreted as emergent dynamics that holographically represents an image of a deeper higher dimensional structure in one extra space and one extra time ...".

From my point of view , two-time physics is just physics based on the Conformal Group Spin(2,4) = SU(2,2).

Since Danny Ross Lunsford's SO(3,3) is isomorphic to SL(4,R) and since SO(3,3) is based on the real Clifford algebra Cl(3,3) = Cl(4,2) = M(8,R) = real 8x8 matrix algebraand since Cl(4,2) gives the Lie algebra SO(4,2) his SO(3,3) physics is closely related to that of the Conformal Group.

Tony Smith

PS - I should note that Cl(4,2) is M(4,Q) = 4x4 quaternionic matrices, so the main difference between Danny Ross Lunsford's SO(3,3) and the conformal Spin(2,4) is the quaternionic structure in the conformal models.

I've read what Lunsford wrote and in my comment above I pointed out that: "My argument above isn't in Lunsford's paper, and his argument is purely abstract; using 3 time dimensions and 3 spatial dimensions you can unify general relativity and electromagnetism making a prediction."

So I'm not misinterpreting Lunsford's 3 dimensional time, just interpreting it.

I've some evidence why there is a pairing up of space and time dimensions, which is independent from, apparently supplements, Lunsford's analysis.

Hubble in 1929 ignored spacetime when he interpreted the recession of galaxies as a velocity directly proportional to apparent distance, v = HR.

You get an equally physically valid and yet entirely different interpretation of the entire universe if you remember that distance R = ct, so that v = Hct, i.e., velocity is directly proportional to the time past you are observing.

This implies an acceleration, of sorts a variation of velocity with effective time. I've never seen anyone else analyse the Hubble law this way (they all try to incorporate it into a suitable metric of GR then have to add an unexplained, ad hoc small positive cosmological constant to make it fit the evidence).

Treating the Hubble expansion rate v = HR as an acceleration is easy:

a = dv/dt = d(HR)/dt = H*dR/dt + R*dH/dt = H*dR/dt

(because H is constant, dH/dt = 0)

a = H*dR/dt = H*v = H*(HR)

This gives a new, physical way to analyse the big bang: every galaxy of mass m has an effective outward force of F = ma = mRH^2.

It appears you don't see the symmetry of having a separate effective time dimension corresponding to each spatial dimension.

"If you have one clock that runs twice as fast as another clock, you end up with two different coordinatizations of time, that differ by a scale factor, but it's still the same time dimension being measured."

You're missing the point, time dilation (or time scaling) only implies a single time dimension when you have one time that is being scaled to take different values at different speeds or in different gravitational fields.

If you have 3 different times coming out, then that implies 3 different time dimensions. How else do you account for 3 different time scaling factors? You will need to scale all three times, you can't scale them all to the same value.

Similarly, you could use your fruitless argument to replace 3 spatial dimensions by 1 spatial dimension and some scale factors:

"The small differences in the apparent radius of the universe (R = c/H) in different directions are not evidence of the existence of 3 different spatial dimensions, and there is only one spatial dimension in the universe with 3 different scaling factors."

Obviously we've got more evidence for 3 spatial dimensions than that, and more evidence than we have for 3 time dimensions, but all the same, you are simply ignoring evidence for 3 time dimensions. How will your scale factors work?

Consider the issue of where the 3 spatial dimensions might look like one spatial dimension. Suppose that you were blind and invalid and has only one sensory source of information, the automatic readout of an ultrasonic distance-measuring gadget that rotates within a room. As it rotates, it gives a different read out to the distance to the walls it is facing, closer or further away.

Instead of admitting the existence of 3 spatial dimensions, if you are consistent, you would presumably claim that there is only one spatial dimension (equivalent to radial distance), with a scale factor accounting for variations.

We can only see spatial dimensions because things are not radially symmetric every direction we look. If everything was the same in all directions we turned, we would appear to be a world with one spatial dimension only. Radial distance would be the single effective dimension, since there would be no basis upon which to discriminate 3 spatial dimensions if space was identical all around us in every direction.

Having one separate time dimension for each separate spatial dimension does seem to be supported by several pieces of evidence. Lunsford shows that there are failures in the Kaluza-Klein theory and others. The simplest which can unify gravity and electrodynamics has 3 spatial and 3 time dimensions. It also makes a correct prediction!

I really think that if you want to be hostile to innovation, you should either read some of the papers and try to find errors, or else maybe you should direct your hostility towards mainstream string theory which has no evidence but a lot more funding and prestige than alternative work.

Nigel, there is no need to be rude. Mitchell is a thoughtful person. Mitchell, Nigel is correct that I was partly thinking of the three times when I mentioned 6 dimensions for em charge, but even more simply, 6 means 0,-1,+1 twice over, because these are the quantum numbers. And the correspondence between the weird operad heirarchies and string dimensions requires a lot more detail than I put into a two paragraph post, and it is always in this context that I make remarks - any remarks.

I find it amazing that anyone is looking at my work, so I might as well say something myself!

In that theory, time is what it is. Time is both duration and persistence. That is, something either exists or not, regardless of how long it exists. This is where the multi-dimensional aspect of the time-like sector of the base space originates.

Now things can cease to persist via annihilation against the anti-partner. So one needs 2 dimensions to describe persistence as we experience it.

In that theory, the base manifold is ontologically a vacuum. The 4-d world exhibits matter as an expression of the 6-D Weyl vacuum. This cannot be stressed enough. Matter in this theory lives in 4-D space, as unique, calibration invariant constructions out of the 6-D geometry of the vacuum.